**3. Discussion**

In the beginning, it should be noted that the exact formula for the skin friction coefficient given by Equation (15) will be invested here and used to validate the numerical results obtained in [20] by applying the homotopy analysis method (HAM) when the second slip vanishes (i.e., at *μ* = 0). The thermophysical properties of water and nanoparticles are introduced in Table 1. These properties have been implemented to conduct the numerical results in Table 2. In view of these comparisons, it may be concluded that the outputs of [20] need some revisions, especially since the differences between the current exact values and the approximate ones seem to be obvious. Besides, the same values of the physical parameters [20] have been selected to hold these comparisons.

**Table 1.** Properties of water and nanoparticles.


**Table 2.** Comparisons between the numerical results of skin friction coefficient [20] and the present exact values for copper and silver at *μ* = 0 and *λ* = 0.4.


In the presence of the second slip, exact values for the skin friction coefficient for the Ag-water and the Cu-water nanofluids are listed in Table 3 at various values of *φ*, *M*, and *γ* when *λ* = 0.4. The results reveal that the skin friction coefficient for both nanofluids increases with an increase in the volume fraction *φ* and the Hartman number *M*; however, it decreases with the increase in the first slip *γ* and with the decrease in the second slip *μ*. Further, the variation of the skin friction coefficient is depicted in Figure 1 against the porosity parameter *λ* at various values of the solid volume fraction *φ* when *μ* = 0. It is clear from this figure that the skin friction increases with increases in both *λ* and *φ*. However, in [20], it was found that this behavior is different than the current one. This also confirms the conclusion made above that the method applied in [20] needs further improvement. In addition, the results in Figure 2 indicate that the skin friction decreases with increases in both *λ* and *φ* in the presence of the second slip parameter. Therefore, the behavior is changed from increasing in Figure 1 (*μ* = 0) to decreasing in Figure 2 (*μ* = −0.5), which confirms the importance of the second slip in modeling the boundary layer flow of nanofluids.

**Table 3.** Values of skin friction coefficient for copper and silver at various values of *φ*, *M*, *γ* and *μ* at *λ* = 0.4.


**Figure 1.** Effects of *φ* on skin friction coefficient when the second slip vanishes.

**Figure 2.** Effects of *φ* on skin friction coefficient in the presence of second slip.

The effect of the first slip parameter *γ* on the velocity of the nanofluids suspended with five nanoparticles is displayed through Figures 3–5. Figures 3 and 4 show that the velocities of the Ag/Cu/TiO2-water nanofluids satisfy *f* (*η*)|Ag < *f* (*η*)|Cu < *f* (*η*)|TiO2 . Figure 5 indicates that *f* (*η*)|SiO2 ≈ *f* (*η*)|Al2O3 ≈ *f* (*η*)|TiO2 . Therefore, it can be concluded from Figures 3–5 that the Ag-water nanofluid is of lower velocity than any of the four other types. This later conclusion is also observed and confirmed through Figures 6–8 for the effect of the second slip *μ* on the velocity of the present five types of nanofluids.

**Figure 3.** Effect of first slip *γ* on velocity of Cu-water and Ag-water nanofluids.

**Figure 4.** Effect of first slip *γ* on velocity of Cu-water and TiO2-water nanofluids.

**Figure 5.** Effect of first slip *γ* on velocity of SiO2-water, Al2O3-water, and TiO2-water nanofluids.

**Figure 6.** Effect of second slip *μ* on velocity of Cu-water and Ag-water nanofluids.

**Figure 7.** Effect of second slip *μ* on velocity of Cu-water and TiO2-water nanofluids.

**Figure 8.** Effect of second slip *μ* on velocity of SiO2-water, Al2O3-water, and TiO2-water nanofluids.

In the absence of the homogenous reaction (i.e., at *K* = 0), the exact solution for the concentration *g*(*η*) is available and given by Equation (16). In that case, the effects of *Ks* and *Sc* on *g*(*η*) are plotted in Figures 9 and 10, respectively. It is shown that a reduction in the concentration occurs with the strengthening of the heterogenous reaction *Ks* and also with the increase in the Schmidt parameter *Sc*. Moreover, the Ag-water nanofluid is of lower concentration than the Cu-water nanofluid. This is also true for the general case, when both of the homogenous and heterogenous reactions take place in Figure 11, where the NDSolve command in Mathematica 7.0 (Wolfram Research, Champaign, IL, USA) has been used to solve the systems (5) and (12).

**Figure 11.** Effects of *Ks* on *g*.
